Lobachevskii Journal of Mathematics http://ljm.ksu.ru Vol. 27, 2007, 3–13

© K. K. Baishya, S. Eyasmin, and A. A. Shaikh

Abstract. The aim of the present paper is to introduce a type of contact metric manifolds called $\phi$-recurrent generalized $\left(k,\mu \right)$-contact metric manifolds and to study their geometric properties. The existence of such manifolds is ensured by a non-trivial example.

1. Introduction

In 1995 Blair, Koufogiorgos, and Papantoniou [5] introduced the notion of $\left(k,\mu \right)$-contact metric manifolds, where $k$ and $\mu$ are real constants, and a full classification of such manifolds was given by E. Boeckx [6]. Assuming $k$, $\mu$ be smooth functions, T. Koufogiorgos and C. Tsichlias introduced the notion of generalized $\left(k,\mu \right)$-contact metric manifolds and gave several examples [8].

The notion of local symmetry of a Riemannian manifold has been weakened by many authors in several ways to a different extent. As a weaker version of local symmetry, T. Takahashi [9] introduced the notion of local $\phi$-symmetry on a Sasakian manifold. Recently, Shaikh etl. [2] studied the locally $\phi$-symmetric $\left(k,\mu \right)$-contact metric manifolds and proved that such a manifold exists whereas a locally $\phi$-symmetric generalized $\left(k,\mu \right)$-contact metric manifold does not exist. Extending the notion of local $\phi$-symmetry, in the present paper we introduce the notion of locally $\phi$-recurrent $\left(k,\mu \right)$-contact metric manifolds and locally $\phi$-recurrent generalized $\left(k,\mu \right)$-contact metric manifolds. In [8], the authors proved that the generalized $\left(k,\mu \right)$-contact metric manifolds exist only for dimension 3 and hence we confined ourselves to the study of 3-dimensional generalized $\left(k,\mu \right)$-contact metric manifolds. The $\left(k,\mu \right)$-contact metric manifold is of our special interest as it contains both the Sasakian and non-Sasakian cases. The paper is organized as follows. Section 2 is concerned with contact metric manifolds. Section 3 deals with $\left(k,\mu \right)$-contact metric manifolds and section 4 is the discussion of generalized $\left(k,\mu \right)$-contact metric manifolds. In section 5 we study locally $\phi$-recurrent $\left(k,\mu \right)$-contact metric manifolds. Section 6 is devoted to the study of locally $\phi$-recurrent generalized $\left(k,\mu \right)$-contact metric manifolds and proved that such a manifold is either flat or an $\eta$-Einstein manifold. Finally, we construct an example of a locally $\phi$-recurrent generalized $\left(k,\mu \right)$-contact metric manifold which is neither locally symmetric nor locally $\phi$-symmetric.

2. Contact metric manifolds

A contact manifold is a $C^{\infty }$ manifold $M^{2n+1}$ equipped with a global 1-form $\eta$ such that $\eta \wedge {\left(d\eta \right)}^n\ne 0$ everywhere on $M^{2n+1}$. Given a contact form $\eta$ it is well known that there exists a unique vector field $\xi$, called the characteristic vector field of $\eta$, such that $\eta \left(\xi \right)$=1 and $d\eta \left(X,\xi \right)=0$ for every vector field X on $M^{2n+1}$. A Riemannian metric is said to be associated metric if there exists a tensor field $\phi$ of type (1,1) such that

for all vector fields X,Y on $M^{2n+1}$. Then the structure $\left(\phi ,\xi ,\eta ,g\right)$ on $M^{2n+1}$ is called a contact metric structure and the manifold $M^{2n+1}$ equipped with such structure is called a contact metric manifold [3].

Given a contact metric manifold $M^{2n+1}$($\phi$, $\xi$, $\eta$, $g$) we define a (1, 1) tensor field $h$ by $h=\frac{1}{2}{\pounds }_{\xi }\phi$, where $\pounds$ denotes the Lie differentiation. Then $h$ is symmetric and satisfies $h\phi =-\phi h$. Thus, if $\lambda$ is an eigenvalue of $h$ with eigenvector X, $-\lambda$ is also an eigenvalue with eigenvector $\phi X$. Also we have $Tr.h=Tr.\phi h=0$ and $h\xi =0$. Moreover, if $\nabla$ denotes the Riemannian connection of $g$, then the following relation holds:

The vector field $\xi$ is a Killing vector with respect to $g$ if and only if $h=0$. A contact metric manifold $M^{2n+1}\left(\phi ,\xi ,\eta ,g\right)$ for which $\xi$ is a Killing vector is said to be a $K$-contact manifold. A contact structure on $M^{2n+1}$ gives rise to an almost complex structure on the product $M^{2n+1}\times R$. If this almost complex structure is integrable, the contact metric manifold is said to be Sasakian. Equivalently, a contact metric manifold is Sasakian if and only if the relation

holds for all $X,Y$, where $R$ denotes the curvature tensor of the manifold. We shall now state a result which will be used later on.

3. $\left(k,\mu \right)$-Contact metric manifolds

For a contact metric manifold $M^{2n+1}\left(\phi ,\xi ,\eta ,g\right)$, the $\left(k,\mu \right)$-nullity distribution is

for any $X,Y\in T_{p}M$, $k$, $\mu$ are real numbers. Hence, if the characteristic vector field $\xi$ belongs to the $\left(k,\mu \right)$-nullity distribution, then we have

Thus a contact metric manifold satisfying relation (3.1) is called a $\left(k,\mu \right)$-contact metric manifold [5]. In particular, if $\mu =0$, then the notion of $\left(k,\mu \right)$-nullity distribution reduces to the notion of $k$-nullity distribution, introduced by S. Tanno [7]. A $\left(k,\mu \right)$-contact metric manifold is Sasakian if and only if $k=1$. In a $\left(k,\mu \right)$-contact metric manifold the following relations hold ([1], [5]):

where $S$ is the Ricci tensor of type (0, 2), $Q$ is the Ricci-operator, i.e., $g\left(QX,Y\right)=S\left(X,Y\right)$ and $r$ is the scalar curvature of the manifold. From (2.4), it follows that

4. Generalized $\left(k,\mu \right)$-contact metric manifolds

A generalized $\left(k,\mu \right)$-contact metric manifold $M^3\left(\phi ,\xi ,\eta ,g\right)$ is a $\left(k,\mu \right)$-contact metric manifold in which $k$ and $\mu$ are smooth functions on $M$. In [8] the authors proved that a generalized $\left(k,\mu \right)$-contact metric manifold does not exist for dimension greater than three. Hence the generalized $\left(k,\mu \right)$-contact metric manifold exists for dimension three and several examples are given in [8]. In a generalized $\left(k,\mu \right)$-contact metric manifold $M^3\left(\phi ,\xi ,\eta ,g\right)$, the relations (3.2), (3.5)-(3.11) hold ([2], [8]) and also the following relations hold :

5. Locally $\phi$-recurrent $\left(k,\mu \right)$-contact metric manifolds

If the 1-form $A$ vanishes identically and the vector fields $X,Y,Z,W$ are orthogonal to $\xi$, then the manifold reduces to a locally $\phi$-symmetric manifold in the sense of Takahashi.

Proof. In a locally $\phi$-recurrent $\left(k,\mu \right)$-contact metric manifold the relation (5.2) holds. Then, by virtue of (2.2), we obtain

Taking the inner product on both sides of (5.3) by any vector field $U$, we get

Let $\left\{e_i:i=1,2,\dots ,2n+1\right\}$ be an orthonormal basis of the tangent space at any point of the manifold. Then setting $X=U=e_i$ in (5.4) and taking summation over $i$, $1\le i\le 2n+1,$ we get

Plugging $Z$ by $\xi$ in (5.5) we obtain, by virtue of the skew-symmetry property of the curvature tensor, that

In view of (3.7) and (3.12), (5.6) reduces to

Setting $Y=\xi$ in (5.7) we get by virtue of (2.2) and (3.7) that

which yields either $k=0$, or $A\left(W\right)=0$ for all vector fields $W$ tangent to $M$.

Again, changing $W$, $X$, $Y$ cyclically in (5.3) and then adding the results, we obtain

which, by virtue of Bianchi identity, yields

In view of (3.6), (9) reduces to

Setting $Y=Z=e_i$ in (5.10) and taking summation over $i$, $1\le i\le 2n+1,$ we get

Substituting $X$ by $\xi$ in (12), we have

If $k=0$, then (5.13) yields either $\mu =0$, or $A\left(hW\right)=0$. Thus for $k=0=\mu$, (3.1) reduces to $R\left(X,Y\right)\xi =0$ for all $X$, $Y$ and hence, by virtue of Lemma 2.1, the manifold under consideration is locally isometric to the Riemannian product $E^{n+1}\left(0\right)\times S^n\left(4\right)$ and, for $n=1$, the manifold is a flat contact metric manifold.

Again, for $k=0$ and $A\left(hW\right)=0$, we have $A\left(W\right)=A\left(\xi \right)\eta \left(W\right)$, which can be written as $A\left(W\right)\eta \left(\xi \right)=A\left(\xi \right)\eta \left(W\right)$. This implies that the vector field $\xi$ and $\rho$ associated to the 1-form $A$ are codirectional. Finally, if $A\left(W\right)=0$, (for $k\ne 0$) for all $W$, then (5.2) reduces to (5.1) and hence the manifold under consideration is locally $\phi$-symmetric in the sense of Takahashi. This proves the theorem.

6. Locally $\phi$-recurrent generalized $\left(k,\mu \right)$-contact metric manifolds

In particular, if $A$ vanishes, then a generalized $\left(k,\mu \right)$-contact metric manifold is said to be a locally $\phi$-symmetric manifold.

Proof. Proceeding similarly to the proof of Theorem 5.1, we can easily show that in a generalized $\left(k,\mu \right)$-contact metric manifold the relation (5.6) holds, and hence in view of (3.7) and (3.12), we obtain

Using (1) in (5.6) we get

Replacing $Y$ by $\phi Y$ in (2) we have

By virtue of (2.2) and (3.11), it follows from (6.3) that

Again, Replacing $W$ by $hW$ in (6.4) and then using (2.2) and (3.2), we obtain

Subtracting (6.5) from (6.4), it follows that either $k=0$, or

The relation (6.6) implies that the manifold under consideration is an $\eta$-Einstein manifold.

If $k=0$, then (6.4) reduces to

Again, if $k=0$, then for $n=1$, (3.9) takes the form

which yields by setting $Y=hY$ that

By virtue of (6.8) and (6.9), we obtain from (6.7) that either $\mu =0$, or $hY=-Y+\eta \left(Y\right)\xi$. If $k=0=\mu$, then the manifold is a flat contact metric structure. Again, if $k=0$ and $hY=-Y+\eta \left(Y\right)\xi$, then (6.8) yields

which implies that the manifold is an $\eta$-Einstein manifold. This proves the theorem.

Proof. In a locally $\phi$-recurrent generalized $\left(k,\mu \right)$-contact metric manifold relation (2) holds. Setting $Y=\xi$ in (2), we obtain $A\left(W\right)=-\frac{1}{k}\left(Wk\right)$ for $k\ne 0,$ which implies that $\rho =-\frac{1}{k}gradk$. This proves the theorem.

Proof. In a locally $\phi$-recurrent generalized $\left(k,\mu \right)$-contact metric manifold relation (2) holds. Setting $W=\xi$ in (2) and then using (4.1), we get $A\left(\xi \right)=0,\text{i.e.,}g\left(\xi ,\rho \right)=0$, provided that $k\ne 0$. This proves the theorem.

We now give an example of a locally $\phi$-recurrent generalized $\left(k,\mu \right)$-contact metric manifold.
Example. We consider the 3-dimensional manifold $M=\left\{\left(x,y,z\right)\in R^3:x\ne 0\right\}$, where $\left(x,y,z\right)$ are the standard coordinates in $R^3$. Let $\left\{E_1,E_2,E_3\right\}$ be linearly independent global frame on $M$ given by

Let $g$ be the Riemannian metric defined by

Let $\eta$ be the 1-form defined by $\eta \left(U\right)=g\left(U,E_3\right)$ for any $U\in \chi \left(M\right)$. Let $\phi$ be the (1, 1)-tensor field defined by $\phi E_1=E_2$, $\phi E_2=-E_1$, $\phi E_3=0$. Then using the linearity of $\phi$ and $g$, we have $\eta \left(E_3\right)=1,{\phi }^{2}U=-U+\eta \left(U\right)E_3$ and $g\left(\phi U,\phi W\right)=g\left(U,W\right)-\eta \left(U\right)\eta \left(W\right)$ for any $U,W\in \chi \left(M\right)$. Moreover $h{E}_1=-E_1,h{E}_2=E_2$ and $h{E}_3=0.$ Thus for $E_3=\xi$, $\left(\phi ,\xi ,\eta ,g\right)$ defines a contact metric structure on $M$.

Let $\nabla$ be the Levi-Civita connection with respect to the Riemannian metric $g$ and $R$ be the curvature tensor of $g$. Then we have

Taking $E_3=\xi$ and using Koszul formula for the Riemannian metric $g$, we can easily calculate

From the above it can be easily seen that $\left(\phi ,\xi ,\eta ,g\right)$ is a generalized $\left(k,\mu \right)$-contact metric structure on $M$. Consequently $M^3\left(\phi ,\xi ,\eta ,g\right)$ is a generalized $\left(k,\mu \right)$-contact metric manifold with $k=-\frac{2}{x}$ and $\mu =-\frac{2}{x}$.

Using the above relations, we can easily calculate the non-vanishing components of the curvature tensor as follows :

and the components which can be obtained from these by the symmetry properties. We shall now show that such a generalized $\left(k,\mu \right)$-contact metric manifold is locally $\phi$-recurrent. Since $\left\{E_1,E_2,E_3\right\}$ from a basis of $M^3$, any vector field $X\in \chi \left(M\right)$ can be written as

where $a_i\in R^{+}$ (the set of all positive real nonumbers), $i=1,2,3$. Thus the covariant derivatives of the curvature tensor are given by

This implies that

Let us now consider the non-vanishing 1-form

at any point $p\in M$. From (6.10)–(6.12), it follows that

This implies that the manifold under consideration is a locally$\phi$-recurrent generalized $\left(k,\mu \right)$-contact metric manifold, which is neither locally symmetric nor locally $\phi$-symmetric. This leads to the following:

References

A. A. Shaikh, K. K. Baishya and S. Eyasmin, Department of Mathematics, University of Burdwan, Golapbag, Burdwan-713104, West Bengal, India

E-mail address: aask2003@yahoo.co.in, aask@epatra.com

Received June 9, 2007